TY - JOUR

T1 - Laplace operators on the cone of Radon measures

AU - Kondratiev, Yuri

AU - Lytvynov, Eugene

AU - Vershik, Anatoly

N1 - Publisher Copyright: © 2015 Elsevier Inc.. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - We consider the infinite-dimensional Lie group G which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold X and the commutative multiplicative group of functions on X. The group G naturally acts on the space M(X) of Radon measures on X. We would like to define a Laplace operator associated with a natural representation of G in L2(M(X),μ). Here μ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure μ is not quasi-invariant with respect to the action of the group G. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group G (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on M(X) whose generator is the Laplace operator.

AB - We consider the infinite-dimensional Lie group G which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold X and the commutative multiplicative group of functions on X. The group G naturally acts on the space M(X) of Radon measures on X. We would like to define a Laplace operator associated with a natural representation of G in L2(M(X),μ). Here μ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure μ is not quasi-invariant with respect to the action of the group G. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group G (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on M(X) whose generator is the Laplace operator.

KW - Laplasian

UR - http://www.scopus.com/inward/record.url?scp=84941260290&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2015.06.007

DO - 10.1016/j.jfa.2015.06.007

M3 - Article

AN - SCOPUS:84941260290

VL - 269

SP - 2947

EP - 2976

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 9

ER -